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An acid is a molecule which can easily loose a proton.

Acids (such as the formic acid shown in figure a) often contain the atoms COOH, where the C (carbon atom) has a double bond with one of the O (oxygen atoms). This is shown by a double line on the diagram.

Electrons like oxygen more than they like hydrogen, so there is only a thin electron shell around the hydrogen atom on the left.
When in water the hydrogen nucleus, the proton p+, can therefore easily escape and wander off, causing damage in other molecules.
Meanwhile the hydrogen atom's electron is left behind on the oxygen atom (O-), so it gains an extra negative charge (figure b).

This can all be put into the simple rule:

An acid is a proton donor

In truth this is a bit of an over-simplification because some molecules behave like acids even though they do not contain any hydrogen atoms which can give away protons, such as boron trichloride. Following the work of G.N.Lewis, most chemists now define acids as molecules which can accept an electron-pair.
Nevertheless the simple rule stated above applies in most cases met in organic chemistry.

Why are acids important in the History of the Universe?
Acids were essential in history for several reasons:

The amino acids which are the building blocks of proteins contain acid.
The weathering of rock, which is vital in the geological cycle, depends in part upon the fact that rain is a weak acid, carbonic acid.
And the electrical forces between electrons left behind after protons are lost by phosphate drives ATP which is life's major energy store.

What is the opposite of an acid?